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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 148, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/148\hfil Dirichlet problem]
{Dirichlet problem for a second order singular
differential equation}
\author[W. Zhou\hfil EJDE-2006/148\hfilneg]
{Wenshu Zhou}
\address{Wenshu Zhou \newline
Department of Mathematics, Jilin University, Changchun
130012, China}
\email{wolfzws@163.com}
\thanks{Submitted July 31, 2006. Published December 5, 2006.}
\thanks{Supported by grants 10626056 from Tianyuan Youth Foundation
and 420010302318 \hfill\break\indent
from Young Teachers Foundation of Jilin University}
\subjclass[2000]{34B15}
\keywords{Singular differential equation; positive solution; existence}
\begin{abstract}
This article concerns the existence of positive solutions to
the Dirichlet problem for a second order singular
differential equation. To prove existence, we use the classical
method of elliptic regularization.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
We study the existence of positive solutions for the second order singular differential equation
\begin{equation}
u''+\lambda\frac{u'}{t-1}-\gamma \frac{|u'|^2}{u}+f(t)=0,
\quad 00$, $f(t) \in C^1[0,1]$ and
$f(t)>0$ on $[0,1]$.
It is well known that boundary value problems for singular
ordinary differential equations arise in the field of
gas dynamics, flow mechanics, theory of boundary layer, and so on.
In recent years, singular second order ordinary differential
equations with dependence on the first order derivative have been
studied extensively, see for example \cite{a1, b4, j1, o1, o2,
s1, t1, w1} and references therein where some general existence
results were obtained. We point out that the case considered here
is not in their considerations since it does not satisfy some
sufficient conditions of those papers. Our considerations were
motivated by the model, which arises in the studies of a
degenerate parabolic equation (see for instance \cite{b1, b2,b3}),
considered by Bertsch and Ughi \cite{b3} in which they
studied \eqref{e1} with $\lambda=0$ and $f\equiv1$ and the
boundary boundary conditions: $ u(1)=u'(0)=0$. By ordinary
differential equation theories, they obtained a decreasing
positive solution. However, it is easy to see from the boundary
conditions \eqref{e2} that any positive solution to the Dirichlet
problem for \eqref{e1} must not be decreasing. Recently, in
\cite{z1} the authors studied the Dirichlet problem for \eqref{e1}
with $\lambda=0$, and proved that if $\gamma>0$, then the problem
has a positive solution $u$; moreover, if $\gamma>\frac{1}{2}$,
then $u$ satisfies also $u'(1)=u'(0)=0$. Note that the equation
considered here is more general since it is also singular at $t=1$
for $\lambda\neq0$. Thus the existence result obtained here is not
a simple extension of \cite{b3, z1}.
We say $u \in C^2(0,1) \cap C[0,1]$ is a solution to the
Dirichlet problem \eqref{e1}, \eqref{e2} if it is positive in
$(0,1)$ and satisfies \eqref{e1} and \eqref{e2}.
The main purpose of this paper is to prove the following theorem.
\begin{theorem}\label{thm1}
Let $\lambda>-1, \gamma>\frac{1}{2}(1+\lambda)$,
$f(t) \in C^1[0,1]$ and $f>0$ on $[0,1]$. Then the Dirichlet problem
\eqref{e1}, \eqref{e2} has a solution $u$. Moreover, $u$ satisfies
$u'(1)=0$. If in addition we assume that $\lambda$ is
non-negative, then $u$ satisfies also $u'(0)=0$.
\end{theorem}
\section{Proof of Theorem \ref{thm1}}
We will use the classical method of elliptic regularization to
prove Theorem \ref{thm1}. For this, we consider the following
regularized problem:
\begin{gather*}
u''+\lambda\frac{u'}{t-1-\varepsilon^{1/2}}-\gamma
\frac{|u'|^2\mathop{\rm sgn}{}_\varepsilon(u)}{I_\varepsilon(u)}+f(t)=0,
\quad 0-1$ and
using \eqref{e5}, we have
\begin{align*}
0 \geqslant& \varepsilon^{\lambda/2}u_\varepsilon'(1)
-\frac{A\varepsilon^{(1+\lambda)/2}}{1+\lambda} \\
\geqslant& (1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'(t)
-\frac{ A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda}\\
\geqslant&(1+\varepsilon^{1/2})^{\lambda}u_\varepsilon'(0)
- \frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda}\\
\geqslant & - \frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda},\quad
t \in [0,1],
\end{align*}
and hence
\[
\frac{A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda} \geqslant
(1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'(t)
\geqslant -\frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda},
\quad t \in [0,1].
\]
This completes the proof of Lemma \ref{lem2.1}.
\end{proof}
Obviously, we have
\begin{gather}
-u_\varepsilon''-\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}}+
\gamma \frac{|u_\varepsilon'|^2}{u_\varepsilon}-\min_{[0,1]}f
\geqslant 0,\quad t \in (0,1),\label{e6}\\
-u_\varepsilon''-\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}}+
\gamma \frac{|u_\varepsilon'|^2}{u_\varepsilon} -
\max_{[0,1]}f\leqslant 0,\quad t \in (0,1).\label{e7}
\end{gather}
To obtain uniform bounds of $u_\varepsilon$,
the following comparison theorem will be proved to be very useful.
\begin{proposition}\label{prop1}
Let $u_i \in C^2(0,1)\cap C[0,1]$ and
$u_i>0$ on $[0,1] (i=1,2)$. If $u_2 \geqslant u_1$ for $t=0,1$, and
\begin{gather}
-u_2''-\eta\frac{u_2'}{t-1-\rho}+\varrho\frac{|u_2'|^2}{u_2}-\theta
\geqslant 0,\quad t \in (0,1),\label{e8}\\
-u_1''-\eta\frac{u_1'}{t-1-\rho}+\varrho\frac{|u_1'|^2}{u_1}-\theta
\leqslant 0,\quad t \in (0,1),\label{e9}
\end{gather}
where $\rho, \varrho, \theta>0, \eta \in \mathbb{R}$, then
$ u_2(t)\geqslant u_1(t)$, $t \in [0,1]$.
\end{proposition}
\begin{proof} From \eqref{e8} and \eqref{e9}, we have
\begin{gather*}
\Big(\frac{u_2^{1-\varrho}}{1-\varrho}\Big)''
+\frac{\eta}{t-1-\rho}\Big(\frac{u_2^{1-\varrho}}{1-\varrho}\Big)'
\leqslant-\frac{\theta}{u_2^{\varrho}},
\quad (\varrho \neq 1)\\
\Big({\rm ln} (u_2)\Big)''
+\frac{\eta}{t-1-\rho}\Big({\rm ln} (u_2)\Big)'
\leqslant-\frac{\theta}{u_2},
\quad (\varrho =1)
\end{gather*}
and
\begin{gather*}
\Big(\frac{u_1^{1-\varrho}}{1-\varrho}\Big)''
+\frac{\eta}{t-1-\rho}\Big(\frac{u_1^{1-\varrho}}{1-\varrho}\Big)'
\geqslant-\frac{\theta}{u_1^{\varrho}},
\quad (\varrho \neq 1)
\\
\Big({\rm ln} (u_1)\Big)''
+\frac{\eta}{t-1-\rho}\Big({\rm ln} (u_1)\Big)'
\geqslant-\frac{\theta}{u_1}.
\quad (\varrho =1)
\end{gather*}
Combining the above inequalities, we obtain
\begin{equation}
w''+\frac{\eta}{t-1-\rho} w'
\leqslant \theta
\Big(\frac{1}{u_1^{\varrho}}-\frac{1}{u_2^{\varrho}}\Big),\quad 0\frac{1}{2}(1+\lambda)$, we find that
\[
-v_{\varepsilon}''-\lambda\frac{v_{\varepsilon}'}{t-1-\varepsilon^{1/2}}+ \gamma
\frac{|v_{\varepsilon}'|^2}{v_{\varepsilon}}-\max_{[0,1]}f
\geqslant 0,\quad 00$ in $(0, 1)$, and
$\lim_{t\to 1^-}u(t)=0$.
Define $u(1)=0$. Thus $u$ is a solution to the Dirichlet
problem \eqref{e1}, \eqref{e2},
and it follows from \eqref{e15} that $u'(1)=0$.
It remains to show that for $\lambda \geqslant 0$, $u $
satisfies $u'(0)=0$. Let
$h_{\varepsilon_j}=C(t+\varepsilon_j^{1/2})^2$, where $C \geqslant
\max\Big\{1, \frac{\max_{[0,1]}f }{2(2\gamma -1)}\Big\}$. Noticing
$\lambda \geqslant 0$ and $\gamma>\frac{1}{2}(1+\lambda)$, we have
\begin{align*}
&-h_{\varepsilon_j}''-\lambda\frac{h_{\varepsilon_j}'}{t-1-\varepsilon_j^{1/2}}+
\gamma
\frac{|h_{\varepsilon_j}'|^2}{h_{\varepsilon_j}}-\max_{[0,1]}f \\
&=2C(2\gamma -1)-2C\lambda\frac{t+\varepsilon_j^{1/2}}{t-1-\varepsilon_j^{1/2}}-\max_{[0,1]}f\\
&\geqslant 2C(2\gamma -1)-\max_{[0,1]}f
\geqslant 0,\quad 0